Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C(·) and the associated sine function S(·) are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(·). Among them are: (1) C(·) is strongly (C, 1)-mean stable (or (C, 2)-mean stable, or Abel-mean stable) if and only if 0 ∈ ρ(A) ∪ σc(A); (2) C(·) is uniformly (C, 2)-mean stable if and only if S(·) is uniformly (C, 1)-mean stable, if and only if t 0 S(s)ds = O(t) (t →∞), if and only if supt>0 t 0 S(s)ds <∞, if and only if C(·) is uniformly Abel-mean stable, if and only if S(·) is uniformly Abel-mean stable, if and only if 0 ∈ ρ(A). © 2006 Elsevier Inc. All rights reserved